2016/10/02 by Yu Wang, Wang, Yu
Mathematics · #Combinatorics #Connection (principal bundle) #Curvature #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Measure (data warehouse) #Monotonic function #Physics #Point processes and geometric inequalities #Stratification (seeds) #Tangent bundle #Tangent space #math.DG
paper · pdf · doi:10.48550/arxiv.1610.00351
27 pages
openalex publication_date 2016/10/02 · arxiv created 2018/03/18 · arxiv updated 2018/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let A be a connection of a principal bundle P over a Riemannian manifold M, such that its curvature FA∈ Lloc2(M) satisfies the stationarity equation. It is a consequence of the stationarity that θA(x,r)=ecr2r4-n∫Br(x)|FA|2 is monotonically increasing in r, for some c depending only on the local geometry of M. We are interested in the singular set defined by S(A)=\x: limr→ 0θA(x,r)≠ 0\, and its stratification Sk(A)=\x: no tangent measure at x is (k+1)-symmetric\. We then introduce and study the quantitative stratification Skε(A). Roughly speaking, Skε(A) consists of points at which no tangent measure of A is ε-close to being (k+1)-symmetric. In the main Theorem, we show that Skε is k-rectifiable and satisfies the Minkowski volume estimate Vol(Br(Skε)∩ B1)≤ Crn-k. Lastly, we apply the main theorems to the stationary Yang-Mills connections to obtain a rectifiability theorem that extends some previously known results by G. Tian.